Solved on Jan 23, 2024

Solve the rational equation $3x^{2/3} - 4 = 8$ .

STEP 1

Assumptions
1. We have a rational equation of the form $3 x^{\frac{2}{3}} - 4 = 8$ .
2. We need to solve for $x$ .

STEP 2

First, we need to isolate the term with the variable $x$ . To do this, we will add 4 to both sides of the equation to move the constant term to the right side.
$3 x^{\frac{2}{3}} - 4 + 4 = 8 + 4$

STEP 3

Simplify both sides of the equation.
$3 x^{\frac{2}{3}} = 12$

STEP 4

To solve for $x$ , we need to get $x$ by itself. We will divide both sides of the equation by 3.
$\frac{3 x^{\frac{2}{3}}}{3} = \frac{12}{3}$

STEP 5

Simplify both sides of the equation.
$x^{\frac{2}{3}} = 4$

STEP 6

To solve for $x$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}$

STEP 7

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ , so we are left with $x$ to the power of 1, which is just $x$ .
$x = 4^{\frac{3}{2}}$

STEP 8

Now we need to calculate $4^{\frac{3}{2}}$ . The exponent $\frac{3}{2}$ means that we take the square root of 4 and then raise it to the power of 3.
$4^{\frac{3}{2}} = (\sqrt{4})^3$

STEP 9

Calculate the square root of 4.
$\sqrt{4} = 2$

STEP 10

Now raise 2 to the power of 3.
$2^3 = 2 \times 2 \times 2$

STEP 11

Calculate the value of $2^3$ .
$2^3 = 8$

STEP 12

Now we have the solution for $x$ .
$x = 8$
The solution to the rational equation $3 x^{\frac{2}{3}} - 4 = 8$ is $x = 8$ .

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Solve the rational equation $3x^{2/3} - 4 = 8$ .

STEP 1

Assumptions
1. We have a rational equation of the form $3 x^{\frac{2}{3}} - 4 = 8$ .
2. We need to solve for $x$ .

STEP 2

First, we need to isolate the term with the variable $x$ . To do this, we will add 4 to both sides of the equation to move the constant term to the right side.
$3 x^{\frac{2}{3}} - 4 + 4 = 8 + 4$

STEP 3

Simplify both sides of the equation.
$3 x^{\frac{2}{3}} = 12$

STEP 4

To solve for $x$ , we need to get $x$ by itself. We will divide both sides of the equation by 3.
$\frac{3 x^{\frac{2}{3}}}{3} = \frac{12}{3}$

STEP 5

Simplify both sides of the equation.
$x^{\frac{2}{3}} = 4$

STEP 6

To solve for $x$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}$

STEP 7

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ , so we are left with $x$ to the power of 1, which is just $x$ .
$x = 4^{\frac{3}{2}}$

STEP 8

Now we need to calculate $4^{\frac{3}{2}}$ . The exponent $\frac{3}{2}$ means that we take the square root of 4 and then raise it to the power of 3.
$4^{\frac{3}{2}} = (\sqrt{4})^3$

STEP 9

Calculate the square root of 4.
$\sqrt{4} = 2$

STEP 10

Now raise 2 to the power of 3.
$2^3 = 2 \times 2 \times 2$

STEP 11

Calculate the value of $2^3$ .
$2^3 = 8$

STEP 12

Now we have the solution for $x$ .
$x = 8$
The solution to the rational equation $3 x^{\frac{2}{3}} - 4 = 8$ is $x = 8$ .

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Solve the rational equation 3x2/3−4=83x^{2/3} - 4 = 83x2/3−4=8.

STEP 1

Assumptions1. We have a rational equation of the form 3x23−4=83 x^{\frac{2}{3}} - 4 = 83x32​−4=8.2. We need to solve for xxx.

STEP 2

First, we need to isolate the term with the variable xxx. To do this, we will add 4 to both sides of the equation to move the constant term to the right side.3x23−4+4=8+43 x^{\frac{2}{3}} - 4 + 4 = 8 + 43x32​−4+4=8+4

STEP 3

Simplify both sides of the equation.3x23=123 x^{\frac{2}{3}} = 123x32​=12

STEP 4

To solve for xxx, we need to get xxx by itself. We will divide both sides of the equation by 3.3x233=123\frac{3 x^{\frac{2}{3}}}{3} = \frac{12}{3}33x32​​=312​

STEP 5

Simplify both sides of the equation.x23=4x^{\frac{2}{3}} = 4x32​=4

STEP 6

To solve for xxx, we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of 23\frac{2}{3}32​, which is 32\frac{3}{2}23​.(x23)32=432(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}(x32​)23​=423​

STEP 7

When we raise a power to a power, we multiply the exponents. The left side simplifies because 23×32=1\frac{2}{3} \times \frac{3}{2} = 132​×23​=1, so we are left with xxx to the power of 1, which is just xxx.x=432x = 4^{\frac{3}{2}}x=423​

STEP 8

Now we need to calculate 4324^{\frac{3}{2}}423​. The exponent 32\frac{3}{2}23​ means that we take the square root of 4 and then raise it to the power of 3.432=(4)34^{\frac{3}{2}} = (\sqrt{4})^3423​=(4​)3

STEP 9

Calculate the square root of 4.4=2\sqrt{4} = 24​=2

STEP 10

Now raise 2 to the power of 3.23=2×2×22^3 = 2 \times 2 \times 223=2×2×2

STEP 11

Calculate the value of 232^323.23=82^3 = 823=8

STEP 12

Now we have the solution for xxx.x=8x = 8x=8The solution to the rational equation 3x23−4=83 x^{\frac{2}{3}} - 4 = 83x32​−4=8 is x=8x = 8x=8.

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Solve the rational equation $3x^{2/3} - 4 = 8$ .

Assumptions
1. We have a rational equation of the form $3 x^{\frac{2}{3}} - 4 = 8$ .
2. We need to solve for $x$ .

First, we need to isolate the term with the variable $x$ . To do this, we will add 4 to both sides of the equation to move the constant term to the right side.
$3 x^{\frac{2}{3}} - 4 + 4 = 8 + 4$

Simplify both sides of the equation.
$3 x^{\frac{2}{3}} = 12$

To solve for $x$ , we need to get $x$ by itself. We will divide both sides of the equation by 3.
$\frac{3 x^{\frac{2}{3}}}{3} = \frac{12}{3}$

Simplify both sides of the equation.
$x^{\frac{2}{3}} = 4$

To solve for $x$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}$

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ , so we are left with $x$ to the power of 1, which is just $x$ .
$x = 4^{\frac{3}{2}}$

Now we need to calculate $4^{\frac{3}{2}}$ . The exponent $\frac{3}{2}$ means that we take the square root of 4 and then raise it to the power of 3.
$4^{\frac{3}{2}} = (\sqrt{4})^3$

Calculate the square root of 4.
$\sqrt{4} = 2$

Now raise 2 to the power of 3.
$2^3 = 2 \times 2 \times 2$

Calculate the value of $2^3$ .
$2^3 = 8$

Now we have the solution for $x$ .
$x = 8$
The solution to the rational equation $3 x^{\frac{2}{3}} - 4 = 8$ is $x = 8$ .